Real-rooted atlas

A038335 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(48 + 27\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1482775
2230425927295625
311059218662410497619683421875
453084161194393610077696377913653144131640625
5254803968716636160806215680453496320127545840143489072373046875
612230590464412782428165804752896043535646720183666009604132485216387420489177978515625
758706834227223115815976963900793946112365699432448020570593075206942575162881301732843041046035320313348388671875
828179280429056126806761930752249650812551168280857164120064197477693521920888649620848642499327058636840167756299522824295364811001129150390625
91352605460594688684756514426060815407021574586368202217158166446081706207272029388895974159051653123599030964436992867623536069632122009559759792762559748498775084686279296875
1064925062108545024365203474360565760924421294475182080138663194171277312013649658176235110409213519268958699524318837157324390401388197657711411202928229434235008036602867927937602058911320946495631351470947265625
11311640298121016115219282743446237872128542327159425440153609151770815304302592010295742167217340416081078969566836555776456069203813455626241832420908179062784051536838042536140809663157132975526401087105177459746725559060566555523422351360321044921875
12149587343098087735296100971456591209221324831238044382905352847365857133321794753658880741293436039648509952066716409243568365895684378264356609174011904211094888622228032716874213046781252042752018553261695313010688031308629110840705536320201888633598124815009463529699912131676352024078369140625
13718019246870821129420852505157427428795088896177204906317572183425024365485119279992628314112513963448987489633566720520387992099833253986304390290994074874940489728219538684167117154025472926178823830025493544962894308824468829667328065121948550548667514889990298925368261493769365905242532745150440525551530189762672375726401805877685546875
143446492384979941421219842714112753171703869210624992347475378404227180134422327818196014095111553024345383437719593033756835843885563674345421629764403232784443502289495001137152210757136800432467864453121037320282689628552770355238899510600861070728888321094048735649217614249984223782695928249057460224314694416149100237053442723317062828752051424109418989131512359209178179480135440826416015625
Technical details
Node id
n-cdff05d2922e65c0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f3c17ca3ebbc301e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:                1;
  1:               48,               27;
  2:             2304,             2592,               729;
  3:           110592,           186624,            104976,             19683;
  4:          5308416,         11943936,          10077696,           3779136,            531441;
  5:        254803968,        716636160,         806215680,         453496320,         127545840,         14348907;
  6:      12230590464,      41278242816,       58047528960,       43535646720,       18366600960,       4132485216,        387420489;
  7:     587068342272,    2311581597696,     3900793946112,     3656994324480,     2057059307520,     694257516288,     130173284304,     10460353203;
  8:   28179280429056,  126806761930752,   249650812551168,   280857164120064,   197477693521920,   88864962084864,   24993270586368,   4016775629952,    282429536481;
  9: 1352605460594688, 6847565144260608, 15407021574586368, 20221715816644608, 17062072720293888, 9597415905165312, 3599030964436992, 867623536069632, 122009559759792, 7625597484987;
%F Formula
The row polynomials satisfy P(n,x) = (48+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(48 + 27*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (48 + 27*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.